By multiplying by the electrostatic potential and integrating, we have recast the differential equation via a weak formulation using the variational method[1]}, [2]}, [3]}, [4]}. Instead of solving for the exact function \(\phi \)  or \(\delta f\)  that satisfies Poisson's equation, we can simply approximate \(\phi \)  and \(\delta f\)  with a suitable function and focus on the dispersion relation itself. Typically, when the Laplacian is kept, the differential equation is put into the weak formulation by integrating the Laplacian term by parts; this technique is well established in other fields such as finite element analysis [5]}.
